Nelson-Aalen Estimator

Definition

Nelson-Aalen Estimatorfor cumulative hazard:

$\hat{H} (t) = \sum_{t_{i} \leq t} \frac{d _{i}}{Y _{i}}$

where:

The Nelson-Aalen estimator calculates cumulative hazard by summing instantaneous hazard rates at each event time.

Step-by-step Intuition:

At time $t_{1}$ : Out of $Y_{1}$ people at risk, $d_{1}$ fail. $h (t_{1}) = \frac{d _{1}}{Y _{1}}$
At time $t_{2}$ : Out of $Y_{2}$ people at risk, $d_{2}$ fail. $h (t_{2}) = \frac{d _{2}}{Y _{2}}$

Cumulative hazard up to $t_{2}$ : $H (t_{2}) = h (t_{1}) + h (t_{2}) = \frac{d _{1}}{Y _{1}} + \frac{d _{2}}{Y _{2}}$

The variance of the Nelson-Aalen estimator:

$Var [\hat{H} (t)] = \sum_{t_{i} \leq t} \frac{d _{i}}{Y _{i}^{2}}$

$\hat{S}_{K M} (t) \approx exp (- \hat{H}_{N A} (t))$